It would be the first time the BBC has shared resources in this way. Local papers would have free access to video content for their websites, access to the BBC's school of journalism and access to iPlayer technology.
Burnham is hosting an industry summit on Tuesday. The summit has been arranged to discuss ways of helping local papers deal with increasing financial pressures.
The summit will be attended by Helen Boaden, the BBC's head of news, and David Holdsworth, controller of the English regions, as well as other industry executives and the National Union of Journalists.
A similar deal was agreed with ITV which allows the broadcaster to access resources to support its regional news services.
The new deal is expected to be modelled on that arrangement. National newspapers, including the Daily Telegraph, and local TV stations such as Channel M, the Manchester-based operation owned by Guardian Media Group, the Guardian's parent company, could also benefit from the deal.